We use N -body simulations to study the evolution of the orbital eccentricities of stars deposited near ( 0.05 pc) the Milky Way massive black hole (MBH), starting from initial conditions motivated by two competing models for their origin: formation in a disk followed by inward migration; and exchange interactions involving a binary star. The first model predicts modest eccentricities, lower than those observed in the S-star cluster, while the second model predicts higher eccentricities than observed. The N -body simulations include a dense cluster of 10M ⊙ stellar black holes (SBHs), expected to accumulate near the MBH by mass segregation. Perturbations from the SBHs tend to randomize the stellar orbits, partially erasing the dynamical signatures of their origin. The eccentricities of the initially highly eccentric stars evolve, in 20 Myr (the S-star lifespan), to a distribution that is consistent at the ∼ 95% level with the observed eccentricity distribution. In contrast, the eccentricities of the initially more circular orbits fail to evolve to the observed values in 20 Myr, arguing against the disk migration scenario. We find that 20%-30% of the S-stars are tidally disrupted by the MBH over their lifetimes, and that the S-stars are not likely to be ejected as hypervelocity stars outside the central 0.05 pc by close encounters with stellar black holes.

Compact object clusters are likely to exist in the centres of some galaxies because of mass segregation. The high densities and velocities reached in them need a better understanding. The formation of binaries and their subsequent merging by gravitational radiation emission are important to the evolution of such clusters. We address the evolution of such a system in a relativistic regime. The recurrent mergers at high velocities create an object with a mass much larger than the average. For this purpose we modified the direct Nbody6++ code to include post‐Newtonian effects on the force during two‐body encounters. We adjusted the equations of motion to include for the first time the effects of both periastron shift and energy loss by emission of gravitational waves, and so to study the eventual decay and merger of radiating binaries. The method employed allows us to give here an accurate post‐Newtonian description of the formation of a runaway compact object by successive mergers with surrounding particles, as well as the distribution of characteristic eccentricities in the events. This study should be envisaged as a first step towards a detailed, accurate study of possible gravitational wave sources, thanks to the combination of the direct Nbody numerical tool with the implementation of post‐Newtonian terms.

Resonant relaxation (RR) is a rapid relaxation process that operates in the nearly-Keplerian potential near a massive black hole (MBH). RR dominates the dynamics of compact remnants that inspiral into a MBH and emit gravitational waves (extreme mass ratio inspiral events, EMRIs). RR can either increase the EMRI rate, or strongly suppress it, depending on its still poorly-determined efficiency. We use small-scale Newtonian Nbody simulations to measure the RR efficiency and to explore its possible dependence on the stellar number density profile around the MBH, and the mass-ratio between the MBH and a star (a single-mass stellar population is assumed). We develop an efficient and robust procedure for detecting and measuring RR in N -body simulations. We present a suite of simulations with a range of stellar density profiles and mass-ratios, and measure the mean RR efficiency in the near-Keplerian limit. We do not find a statistically significant dependence on the density profile or the mass-ratio. Our numerical determination of the RR efficiency in the Newtonian, single-mass population approximations, suggests that RR will likely enhance the EMRI rate by a factor of a few over the rates predicted assuming only slow stochastic two-body relaxation.

Orbital energy relaxation around a massive black hole (MBH) plays a key role in establishing the stellar dynamical state of galactic nuclei, and the nature of close stellar interactions with the MBH. The standard description of this process as diffusion in phase space provides a perturbative 2nd-order solution in the weak two-body interaction limit. We carry out a suite of N -body simulations, and find that this solution fails to describe the non-Gaussian short timescale evolution of the energy, which is strongly influenced by extreme events (a "heavy-tailed" distribution with diverging moments) even in the weak limit, and is thus difficult to characterize and measure reliably. We address this problem by deriving a non-perturbative solution for energy relaxation as an anomalous diffusion process driven by two-body interactions, and by developing a robust estimation technique to measure it in N -body simulations. These make it possible to analyze and fully model our numerical results, and thus empirically validate in detail, for the first time, this theoretical framework for describing energy relaxation around an MBH on all timescales. We derive the relation between the energy diffusion time, t E , and the time for a small density perturbation to return to steady state, t r , in a relaxed, single mass n(r) ∝ r −7/4 cusp around a MBH. We constrain the modest contribution from strong stellar encounters, and measure with high precision that of the weakest encounters, thereby determining the value of the Coulomb logarithm, and providing a robust analytical estimate for t E in a finite nuclear stellar cusp. We find that t r ≃ 10t E ≃ (5/32)Q 2 P h /N h log Q, where Q = M • /M ⋆ is the MBH to star mass ratio, the orbital period P h and number of stars N h are evaluated at the energy scale corresponding to the MBH's sphere of influence, E h = σ 2 ∞ , where σ ∞ is the velocity dispersion far from the MBH. We conclude, scaling σ ∞ by the observed cosmic M • /σ ∞ correlation, that stellar cusps around lower-mass MBHs (M • 10 7 M ⊙ ), which have evolved passively over a Hubble time, should be dynamically relaxed. We briefly consider the implications of anomalous energy diffusion for orbital perturbations of stars observed very near the Galactic MBH.

In the braneworld scenario the four dimensional effective Einstein equation has extra source terms, which arise from the embedding of the 3-brane in the bulk. These non-local effects, generated by the free gravitational field of the bulk, may provide an explanation for the dynamics of the neutral hydrogen clouds at large distances from the galactic center, which is usually explained by postulating the existence of the dark matter. In the present paper we consider the asymptotic behavior of the galactic rotation curves in the brane world models, and we compare the theoretical results with observations of both High Surface Brightness and Low Surface Brightness galaxies. For the chosen sample of galaxies we determine first the baryonic parameters by fitting the photometric data to the adopted galaxy model; then we test the hypothesis of the Weyl fluid acting as dark matter on the chosen sample of spiral galaxies by fitting the tangential velocity equation of the combined baryonic-Weyl model to the rotation curves. We give an analytical expression for the rotational velocity of a test particle on a stable circular orbit in the exterior region to a galaxy, with Weyl fluid contributions included. The model parameter ranges for which the $\chi^2$ test provides agreement (within 1$\sigma$ confidence level) with observations on the velocity fields of the chosen galaxy sample are then determined. There is a good agreement between the theoretical predictions and observations, showing that extra-dimensional models can be effectively used as a viable alternative to the standard dark matter paradigm.Comment: to be published in MNRAS, 17 pages, 31 figures, version including corrections on the proo